1
$\begingroup$

I am trying to implement the forward Monte Carlo algorithm from the paper "A Forward Monte Carlo Method for American Options Pricing" by Daniel Wei-Chung Miao and Yung-Hsin Lee. I am a little bit confused by the following notation:

Under the Pseudo-Critical Prices section, the authors state:

First, consider an American call option. According to Barone-Adesi and Whaley (1987) (BAW), the optimal exercise boundary $S_c^{*}$ for the call option should solve the nonlinear equation at any time $t\in[0,T]$

\begin{equation} \label{eq:2} S_c^{*} = \frac{Q_2(C_e(S_c^{*}) + K)}{Q_2 - (1-C_e^{'}(S_c^{*}))} \end{equation}

where $C_e(S)$ is the European call option price calculated by the Black-Scholes (1973) formula, $K$ is the strike price, together with the notation $Q_2 = \frac{-(n-1)+\sqrt{(n-1)^2 + 4m/k}}{2} > 0$ in which $m = \frac{2r}{\sigma^2}$, $n = \frac{2(r-q)}{\sigma^2}$, and $k = 1 - e^{-r(T-t)}$. Note that this study used more streamlined notations such as $S_e^{*} = S_c^{*}(t)$, $C_e(S) = C_e(S,t)$ when some dependent parameters are not stressed.

Replacing the critical price $S_c^{*}$ on the right-hand side of the equation above with the current stock price $S$ yields a new but closely related function $f_c(\cdot)$. $$\hat{S_c} = f_c(S) = \frac{Q_2(C_e(S) + K)}{Q_2 - (1-C_e^{'}(S))}$$ where $\hat{S_c}$ represents the pseudo-critical price.

In the algorithm I need to compute $\hat{S_c}$ but what I am not understanding is it seems that $Q_2$ will just equal some number but then the formula for $\hat{S_c}$ has $Q_2$ acting as some function which does not make sense since it is just a number. Any suggestions or comments on the matter are greatly appreciated.

$\endgroup$
1
$\begingroup$

To simply answer this question the author is just multiplying the numbers.

$\endgroup$
  • $\begingroup$ Thanks for contributing back by answering your question! $\endgroup$ – LocalVolatility Mar 23 '17 at 23:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.